Push a swing at just the right rhythm and it goes higher and higher on tiny pushes. Every object has a natural note; drive it at THAT frequency and a small force builds a huge motion — a wine glass shatters, a bridge sways, a string sings. Slide the drive frequency across the natural one and watch the amplitude spike.
A driven damped harmonic oscillator has steady-state amplitude A(ω) = 1/√((ω₀²−ω²)²+(γω)²), which peaks near the natural frequency ω₀ and is sharper the lower the damping — the quality factor Q = ω₀/γ measures that sharpness (the peak is ~Q times the static response). At resonance the driving force feeds energy in phase every cycle, so a small periodic push accumulates into a large oscillation. A fail-loud self-check throws unless the amplitude at ω₀ is several times the off-resonance amplitude. ◆ real acoustics/physics, node-verified.
A single-mode linear resonator (Q=20); real objects have many modes and non-linear limits (the glass finally breaks), and true infinite resonance needs zero damping — the peak-at-the-natural-frequency and the Q sharpness are exact.